Are feature maps merged or are they passed on as they are?
tl;dr
It helps to think that the channels dimension of a convolutional layer works like a fully connected layer (i.e. the layer computes the weighted sum over all channels).
For a single pixel...
Let's consider a single pixel (e.g. the top left pixel). This pixel has $C$ different values, where $C$ are the number of channels. In order to produce the result of a single filter, the layer takes the weighted sum of these $C$ pixels. It does this by actually having $C$ weights, multiplying the pixel values with their corresponding weights and summing them together.
Example
Let's say you have an $n \times m \times 10$ tensor as an input to a convolutional layer with $1$ filter and a $3 \times 3$ kernel. For creating its output the layer has $3 \times 3 \times 10 = 90$ weights, i.e. a different $3 \times 3$ kernel for each of the $10$ input channels. To create its output, the layer performs the convolution operation separately on each of the input channels (each with its corresponding weight matrix) and creates this way $10$ feature maps which are summed together.
Now imagine the layer has $20$ filters instead of $1$. Nothing changes, just that the same procedure is done $20$ times with $20$ different sets of weights So the total number of weights in the layer in this case is $3 \times 3 \times 10 \times 20 = 1800$.
To answer your questions...
(1) You have a grayscale image $n \times m \times 1$ and it passes through the first convolution layer (which has $10$ filters). This layer will perform the convolution operation on the input image $10$ times independently and produce $10$ feature maps, i.e. an output tensor of $n \times m \times 10$. Now this in fed into the second convolution layer, which again has $10$ filters. For each filter the layer will perform the convolution operation on the $10$ input feature maps independently and will sum the corresponding pixels together to form a $n \times m \times 1$ feature map. It will perform this procedure $10$ times and generate an output tensor of $n \times m \times 10$.
(2) Yes, each layer actually has $30$ filters, a total of $10$ for each of the R, G, B channels. The layer just sums the results of the R, G, B channels to produce a single feature map. For the second part, if I got it right it's pretty much what you said but it sums the maps instead of averaging them
Notes
I'd recommend checking Stanford's CS231 notes on convolution layers, which explain it in much detail and also have numerical examples to confirm if you've got it right.
You could also check this answer for more details.